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Published in:
2015 | OriginalPaper | Chapter
Traces and Characteristic Classes in Infinite Dimensions
Authors : Yoshiaki Maeda, Steven Rosenberg
Published in: Geometry and Analysis on Manifolds
Publisher: Springer International Publishing
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We survey geometric constructions of characteristic classes associated to certain infinite rank bundles on the loop space
LM
of a manifold
M
. There are two types of classes, which arise from applying either the leading order trace or the Wodzicki residue to the curvature of natural connections on
TLM
, as the curvature forms take values in pseudodifferential operators. The leading order classes lead to a restatement of the
S
1
-index theorem on
LM
, provide generators for the cohomology of loop groups, and for Maps(
S
2
,
M
) are related to Gromov-Witten invariants. The Wodzicki classes have applications to the topology of diffeomorphism groups of certain circle bundles over Kaehler surfaces.